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Rational, Log Canonical, Du Bois Singularities: On the Conjectures of Kollár and Steenbrink

Published online by Cambridge University Press:  04 December 2007

SÁNDOR J. KOVÁCS
SÁNDOR J. KOVÁCS
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. e-mail: [email protected]
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Abstract

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Let X be a proper complex variety with Du Bois singularities. Then H(X,C)→ H(X,${\mathcal O}$) is surjective for all i. This property makes this class of singularities behave well with regard to Kodaira type vanishing theorems. Steenbrink conjectured that rational singularities are Du Bois and Kollár conjectured that log canonical singularities are Du Bois. Kollár also conjectured that under some reasonable extra conditions Du Bois singularities are log canonical. In this article Steenbrink‘s conjecture is proved in its full generality, Kollár‘s first conjecture is proved under some extra conditions and Kollár‘s second conjecture is proved under a set of reasonable conditions, and shown that these conditions cannot be relaxed.

Keywords

singularitiesrationallog canonicalDu Bois.
Type
Research Article
Information
Compositio Mathematica , Volume 118 , Issue 2 , September 1999 , pp. 123 - 133
Copyright
© 1999 Kluwer Academic Publishers